CoolStuffYT

Your question is x*3x+1??

AoPS problem, eh?

Check here, has some hints, I used them for this problem: https://math.stackexchange.com/questions/3647594/single-file-queue-with-people-as-blockers

A perpendicular slope is the negative reciprocal. So for -2, it becomes 1/2 (multiply by -1, and take the reciprocal). The slope is 1/2, and it passes through (-3, 9). Can you solve from here? (Extra hint: Write out point slope form, then fill in what you know.)

This is probably not the most efficient way, but it works.

Let O = (0, 0), D = (0, -8). Then B = (-12, -8) because OD = 8 and OB = 4sqrt(13), so by the Pythagorean Theorem, BD = 12

Then, since AB is a line segment passing through O, we have that A = (12, 8). Use the distance formula from (0, -8) to (12, 8) and get sqrt((12-0)^2+(8-(-8))^2) = sqrt(12^2+16^2) = 20, so AD = 20.

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This means 2^(10x) = 9.10x = log9, so x = log9/10, which is B.

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The circle is x^2+y^2 = 25, so system of equations with 3x+4y-7 = 0. We get (-3, 4) and (117/25, -44/25). Four coords are (0, 0), (-3, 4), (3, 4), (4.68, -1.76). Use Shoelace Theorem (more info https://artofproblemsolving.com/wiki/index.php/Shoelace_Theorem)

Using the Shoelace theorem, we get the area as 21.5.

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Think about it like this: o is a marble and | are dividers into 4 jars. For example we could have o | ooo | o | oo, which is 1, 3, 1, 2 in the four jars.

There are 7+3 things total, which is 10. There are 3 dividers, so the number of orders of this "string" of o's and |'s is 10 choose 3 = 120.

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Parallel lines always have the same slope. In this case, it's -5/4.

Perpendicular lines multiply to -1. In this case, it's 4/5. Note that a simple way to get 4/5 is to reverse -5/4 to -4/5, then multiply by -1 and get rid of the negative sign.

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BY and CZ are angle bisectors, so by the angle bisector theorem, AZ/AC = BZ/BC and CY/BC = AY/AB. Plugging in some values from the givens gets us \(\frac{AZ}{AC} = \frac{8-AZ}{BC}\) and \(\frac{4}{BC} = \frac{6}{8}\), so BC = 16/3. Also note that AC = AY+CY = 6+4 = 10, so we have AZ/10 = 8-AZ/(16/3). Solving for AZ gets us 120/23, and since AB is 8, AB-AZ = BZ so BZ = 64/23.

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Hint: calculate angle AED and BEC first.